CLARK-OCONE FORMULA BY THE S-TRANSFORM ON THE POISSON WHITE NOISE SPACE

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1 Communications on Stochastic Analysis Vol. 6, No. 4 ( Serials Publications CLARK-OCONE FORMULA BY THE S-TRANSFORM ON THE POISSON WHITE NOISE SPACE YUH-JIA LEE*, NICOLAS PRIVAULT, AND HSIN-HUNG SHIH* Abstract. Given ϕ a square-integrable Poisson white noise functionals we show that the Segal-Bargmann (S- transform t Sϕ(P t(g is absolutely continuous with d dt Sϕ(Pt(g = S( te[ tϕ F t] (g for almost all t R, where P t(g = 1 (,t] g for g in the Schwartz space S on R, and t means the Poisson white noise derivative. After integration with respect to t and applying the inverse S-transform, this identity recovers the Clark-Ocone formula for ϕ. 1. Introduction The Clark-Ocone formula allows one to express a given random variable ϕ defined on the classical Wiener space as the sum of its expectation value and an Itô integral with respect to Brownian motion. This formula was first derived by J. M. C. Clark(1970 in [2], and later, generalized to weakly differentiable functionals byd. Ocone(1984in [13]. Suchaformulaisauseful toolin mathematicalfinance, for example, to compute the replicating portfolio of a claim with given maturity (see [1]. Recently, non-gaussian processes and more general Lévy processes have been extensively studied and new applications have been found in financial mathematics, quantum probability, stochastic analysis, etc. (see [1, 10, 12, 14, 16]. It is natural to ask that if the Clark-Ocone formula for Gaussian functionals can be generalized to non-gaussian Lévy white noise functionals. In Hida s white noise theory, the Segal-Bargmann transform (S-transform, for abbrev. plays a key role. There the generalized white noise fuctionals have been defined and studied through their S-transforms (see [6]. In [9], Lee and Shih have shown that the S-transform allows one to connect the Clark-Ocone formula of Brownian functionals with the fundamental theorem of calculus for Segal-Bargmann analytic functions. Applying the integral representation of the S-transform of Lévy white noise functionals in Received ; Communicated by the editors. Article is based on a lecture presented at the International Conference on Stochastic Analysis and Applications, Hammamet, Tunisia, October 10-15, Mathematics Subject Classification. Primary 60H40; Secondary 60H05, 60H07, 60J75. Key words and phrases. Clark-Ocone formula, Poisson white noise, Poisson measure, white noise analysis. * Research supported by the NSC of Taiwan. Research supported by NTU Start-Up Grant M

2 514 YUH-JIA LEE, NICOLAS PRIVAULT, AND HSIN-HUNG SHIH the non-gaussian case, Lee and Shih [7], have developed a theory of the Lévy white noise analysis for case of Lévy processes following Hida s idea and by means of the S-transform, cf. [10] (see also [12]. In this paper we are concerned with the derivation of the Clark-Ocone formula for Poisson white noise functionals. The main result is obtained in Section 4, in which we apply the results of [10] to the derivation of the Clark-Ocone formula for square-integrable Poisson white noise functionals. The main argument can be briefly described as follows. Given g an element of the Schwartz space S on R and P t (g = 1 (,t] g, we prove that the function t Sϕ(P t (g is absolutely continuous, where Sϕ is the Segal-Bargmann (S- transform of the square-integrable Poisson white noise functional ϕ. As a consequence we obtain the following identity Sϕ(g = E[ϕ]+ + d dt Sϕ(P t(gdt, g S, (1.1 by the fundamental theorem of calculus. Comparing with the Clark-Ocone formula (Theorem 4.2, we show that for almost all t R we have d dt Sϕ(P t(g = S ( te[ t ϕ F t ] (g, g S, where t denotes the Poisson white noise derivative. This also shows that we can recover the Clark-Ocone formula for ϕ simply by taking the inverse S-transform for the equation (1.1. We note that the results of this paper can be extended to compound Poisson or more general Lévy white noise functionals. Since the arguments are much more involved, they will be discussed in a forthcoming paper. 2. Elements of Poisson White Noise Analysis In this section we briefly review some notions and basic results of Poisson white noise analysis. The interested reader is referred to [10] for details and generalization to Lévy white noise analysis. Let X = {X(t; t R} be a standard Poisson process with X(0 = 0 almost surely, and E[e i r(x(t X(s ] = exp((t s(e i r 1, r R, 0 s t, (2.1 where i = 1. Let S be the Schwartz space with the dual S of tempered distributions on R. It is well-known that there exists a unique probability measure Λ on (S,B(S with the characteristic functional C on S given by ( + C(η E[e i,η ( ] = exp e i η(t 1 dt, η S, where, is the S -S pairing and dt refers to the Lebesgue measure (see [7]. The probability measure Λ is then called the Poisson white noise measure and (S,B(S,Λ will serve as the underlying probability space. For any η S, the mean and variance of the random variables, η are given by E[, η ] = + η(tdt, and V[, η ] = + η(t 2 dt. (2.2

3 CLARK-OCONE FORMULA OF POISSON FUNCTIONALS 515 For each ρ L 1 L 2 (R,dt, choose a sequence {η n } S so that η n ρ in L 1 L 2 (R,dt under the norm L 1 (R,dt + L 2 (R,dt. Then it follows from (2.2 that {, η n } forms a Cauchy sequence in L 2 (S,Λ. Denote by, ρ the L 2 -limit of {, η n }. When ρ = 1 (s,t], the indicator of (s, t], the characteristic function of,ρ is exactly the same as in (2.1. So the Poisson process X on (S,B(S,Λ can be represented by x, 1 [0,t], if t 0 X(t; x = x, 1 [t,0], if t < 0, x S. Taking the time derivative formally, we get Ẋ(t; x = x(t, x S. From this viewpoint, the elements ofs can be regardedasthe sample paths ofpoissonwhite noise, and thus members of L 2 (S,Λ are also called square-integrable Poisson white noise functionals. For η = η 1 +iη 2 L 1 c L 2 c(r,dt with η 1, η 2 L 1 L 2 (R,dt,, η is defined to be, η 1 +i, η 2, where V c denotes the complexification of a real locally convex space V. Then the above identity in (2.2 also hold for complex-valued random variable, η. Fock-type decomposition. Let B b (R be the class of all bounded Borel subsets E of R. For any E B b (R, let N(E; be the integer-valued function on (S,B(S defined by N(E; x = # {t E : X(t; x X(t ; x = 1}, x S. ThenN(E; isarandomvariablewhichhaspoissondistributionandthelebesgue measurelebonristhecorrespondingintensitymeasure. Thesystemof{N(E; x Leb(E : E B b (R; x S } forms an independent random measure with zero. We denote N(E; x Leb(E by Ñ(E; x. Then for f L2 c (R, the stochastic integral f(tdñ(t has mean zero and the variance f(t 2 dt. We note that as f L 1 c L2 c (R, f(tdñ(t; x = f(tdx(t; x f(tdt [Λ]-almost everywhere x S, where the right-hand side integral with respect to X exists pathwise in the sense of the Lebesgue-Stieltjes integrals. In particular, for b > a, X(b X(a = (b a+ 1 (a,b] (tdñ(t. Notice that, for any E,F B(R, E[Ñ(EÑ(F] = Leb(E F. Denote by I n (g, g ˆL 2 c (Rn, the closed subspace of L 2 c (Rn consisting of all symmetric complex-valued functions in L 2 c(r, the multiple stochastic integral of order n with respect to Ñ g(t 1,...,t n dñ(t1 dñ(t n,

4 516 YUH-JIA LEE, NICOLAS PRIVAULT, AND HSIN-HUNG SHIH which satisfies the isometry In (g 2 L 2 (S,Λ = n! g(t1,...,t n 2 dt 1 dt n. The multiple stochastic integrals with respect to Ñ provide an orthogonal decomposition theorem for the square-integrable Poisson white noise functionals. Theorem 2.1. (Itô [4] Let ϕ be given in L 2 (S,Λ. Then (i there exist uniquely a series of kernel functions φ n ˆL 2 c (Rn, n N {0}, such that ϕ is equal to the orthogonal direct sum I n(φ n. In notation, we write ϕ (φ n. (ii L 2 (S,Λ is isomorphic to the symmetric Fock space F s (L 2 c(r over L 2 c(r by carrying ϕ (φ n into ( 0!φ 0, 1!φ 1,..., n!φ n,... The Segal-Bargmann transform. For ϕ L 2 (S,Λ, say ϕ (φ n, the Segal- Bargmann (or the S- transform of ϕ is a complex-valued functional on L 2 c(r by Sϕ(g ϕ(xe X (g(xλ(dx S = φ n (t 1,...,t n g(t 1 g(t n dt 1 dt n, where E X (g = 1 n! I n(g n is the coherent state functional associated with g. The S-transform is a unitary operator mapping from L 2 (S,Λ onto the Bargmann-Segal-Dwyer space F 1 (L 2 c (R over L 2 c (R,. Moreover, we have ϕ 2 L 2 (S,Λ = 1 n! Dn Sϕ(0 2 L (n (2 (L2(R, c whered isthe FréchetderivativeofSϕand (n L denotesthehilbert-schmidt (H (2 operator norm of a n-linear functional on a Hilbert space H. Theorem 2.2. (Lee-Shih[7] Let ϕ L 2 (S,Λ. Then for any g L 1 c(r L c (R, we have ( E X (g = exp g(t dt (1+g(t, t J X(x where J X (x is the set {t R : X(t; x X(t ; x = 1}. Remark 2.3. Let H be a complex Hilbert space. For r > 0, denote by F r (H the class of analytic functions on H with norm F r (H such that f 2 F r (H = r n called the Bargmann-Segal-Dwyer space. n! Dn f(0 2 < +, L (n [H] (2

5 CLARK-OCONE FORMULA OF POISSON FUNCTIONALS 517 Test and generalized Poisson white noise functionals. Let A = d 2 /dt 2 +(1+t 2 denote the densely defined and self-adjoint operator on L 2 (R,dt and {e n : n N 0 } be eigenfunctions of A with corresponding eigenvalues 2n+2, n N {0}. {e n : n N 0 } forms a complete orthonormal basis (CONS, in abbreviation of L 2 (R,dt. For any p R and η L 2 (R,dt, define η p := A p η L 2 (R,dt and let S p be the completion of the class {η L 2 (R,dt : η p < + } with respect to p -norm. Then S p is a real separable Hilbert space and we have the continuous inclusions for p > q 0: S = lim p>0 S p S p S q L 2 (R,dt S q S p S = lim p>0 S p. We next proceed to construct the spaces of test and generalized functions as follows. For p R and ϕ L 2 (S,Λ, define ϕ 2 1 p = n! Dn Sϕ(0 2 L (n (2 (S p and let L p be the completion of the class {ϕ L 2 (S,Λ : ϕ p < + } with respect to p -norm. Then L p, p R, is a Hilbert space with the inner product, p induced by p -norm. Forp,q R with q p, L q L p and the embedding L q L p is of Hilbert-Schmidt type, whenever q p > 1/2. Let L = lim L p. p>0 Then L is a nuclear space which will serve as the space of test functions and the dual space L of L the space of generalized functions. The members of L are called generalized Poisson white noise functionals. In this way, we obtain a Gel fand triple L L 2 (S,Λ L and have the continuous inclusion: L L p L q L 2 (S,Λ L q L p L = lim p>0 L p, p q > 0. In what follows, the dual pairing of L and L will be denoted by,. For g L 2 c(r, it is easy to see that E X (g p = e (1/2 g 2 p for any p > 0. Hence E X (g L p if and only if g S p,c for p > 0. We define the S-transform for F L by SF(g = F, E X (g, g S c. Annihilation and creation operators. Let F L p and ξ S p,c, p R. The Gâteaux derivative (d/dz z=0 SF( + zξ in the direction ξ is an analytic function on S p,c. In fact, by using the Cauchy integral formula, one can show that (d/dz z=0 SF( +zξ L p 1. Define ( d ξ F = S 1 dz SF( +zξ z=0. Then by the characterization theorem [11], we have ξ F L p 5. It is clear that 2 ξ is continuous from L into itself. Its adjoint operator ξ is then defined from by ξ F, ϕ := F, ξ ϕ for F L and ϕ L. Here, ξ is called the annihilation operator and ξ is called the creation operator. For convenience, we denote the Poisson white noise derivative δt and its adjoint δ t respectively by t and t for t R, where δ t is the Dirac measure concentrated on the point t.

6 518 YUH-JIA LEE, NICOLAS PRIVAULT, AND HSIN-HUNG SHIH Proposition 2.4. ([10] Let ϕ (φ n L p and ξ S p,c for p 0. If p q ln3/6ln2, then ξ ϕ = ξ, φ 1 + n ξ, φ n (,t 2,...,t n dñ(t2 dñ(t n, n=2 where, is the S p,c -S p,c pairing. Moreover, ξ ϕ q 2 ξ p ϕ p. From [10] we have the following proposition. Proposition 2.5. ([10] Let F (f n L p and ξ S c. Then ξ F = Moreover, for q p ln3/6ln2, I n+1 (ξ ˆ f n in L. ξ F q 2 ξ q F p. 3. Fundamental Theorem of Calculus on Bargmann-Segal-Dwyer Spaces First of all, we specify some notations as follows: - For t R and h L 2 c (R we let P t(h = h 1 (,t]. - For g L 1 L (R and x S we set ( γ 1 (g = exp g(t dt, γ 2 (g(x = and γ(g = γ 1 (gγ 2 (g. t J X(x (1+g(t, Theorem 3.1. For any ϕ L 2 (S,Λ and g L 1 L (R, the mapping t R Sϕ(P t (g is absolutely continuous. Moreover, for any < a < b < +, Sϕ(P b (g Sϕ(P a (g = b a d dt Sϕ(P t(gdt. (3.1 Proof. Letg L 1 L (R andϕ L 2 (S,Λbe fixed. We startbyprovingthe absolutecontinuityofthe mappingt R Sϕ(P t (g. Let {[a i,b i ] : i = 1,2,3,...} be any (finite or countably infinite collection of nonoverlapping intervals. Then Sϕ(P bi (g Sϕ(P ai (g i = i ϕ(x(γ(p bi (g(x γ(p ai (g(xλ(dx. (3.2 S

7 CLARK-OCONE FORMULA OF POISSON FUNCTIONALS 519 We denote this sum by N. To estimate the sum N, we first note that γ 1 (P bi (g γ 1 (P ai (g ϕ(x γ 2 (P bi (g(x Λ(dx (3.3 N i S + γ 1 (P ai (g ϕ(x γ 2 (P bi (g(x γ 2 (P ai (g(x Λ(dx. (3.4 i S Let N (1 and N (2 stand separately for the sums in the right hand side of (3.3 and (3.4. Since, for any a < b, γ 1 (P b (g γ 1 (P a (g b a ( g(t dt exp 2 g(t dt and ( γ 2 (P b (g(x exp g(t dt E X (P b (g(x, x S, we see, by the Cauchy-Schwarz inequality, that (1 C(ϕ,g g(t dt, (3.5 N i [a i,b i] where C(ϕ,g is a constant, depending only on ϕ and g, given by ( C(ϕ,g = ϕ L 2 (S,Λ exp 3 g(t dt+ 1 g(t 2 dt. R 2 R For the estimation of N (2, we need the following inequality: Lemma 3.2. Let E, F be two countable subsets of complex numbers such that E F and z F z < +. Then (1+z (1+z (1+ z (1+ z. z F z E z F z E By Lemma 3.2, we see that for any a < b, γ 2 (P b (g(x γ 2 (P a (g(x ( b exp b a ln(1+ g(t dx(t; x ln(1+ g(t dx(t; x exp ( a exp ln(1+ g(t dx(t; x ( 2 ln(1+ g(t dx(t; x. (3.6

8 520 YUH-JIA LEE, NICOLAS PRIVAULT, AND HSIN-HUNG SHIH It follows from (3.6 that ( (2 e g(t dt ϕ(x ln(1+ g(t dx(t; x N S i [a i,b i] ( exp 2 ln(1+ g(t dx(t; x Λ(dx where e g(t dt ϕ L 2 (S,Λ ( ln(1+ g(t dx(t; x S i[a i,b i] ( exp 4 D(ϕ,g ( exp 5 { S 2 ] } 1/2 ln(1+ g(t dx(t; x Λ(dx ( ln(1+ g(t dx(t; x i[a i,b i] 1/2 ln(1+ g(t dx(t; x Λ(dx}, (3.7 D(ϕ,g = e g(t dt ϕ L 2 (S,Λ is a constant depending only only on ϕ and g. Note that for any two non-negative Borel measurable functions p(t, q(t, t R, we have S ( = p(t dx(t; x ( p(te q(t dt ( exp exp By applying this formula to (3.7, we see that ( (2 D(ϕ,g N exp q(t dx(t; x ( (e q(t 1dt Λ(dx. (1+ g(t 5 ln(1+ g(t dt i[a i,b i] ( 1 2 ( (1+ g(t 5 1 dt, 1/2 which tends to zero whenever (b i a i tends to zero. Combining this bound with (3.5 yields the absolute continuity property and ( Clark-Ocone Formula for Poisson White Noise Functionals In [2], Clark proved that a sufficiently well-behaved Fréchet-differentiable Brownian functional can be represented as a stochastic integral in which the integrand has the form of conditional expectations of the differential. We note that this result can be extended to all square integrable Poisson white noise functionals.

9 CLARK-OCONE FORMULA OF POISSON FUNCTIONALS 521 Lemma 4.1. [8] Let F (f n L 2 (S,Λ. The conditional expectation E[F F t ] of F relative to F t, t R, is given by E[F F t ] = I n (1 (,t] n f n, where E n means the product E E for any E B(R. Let f n ˆL 2 c (Rn and ψ (ψ n L. We observe that for almost all t R, t I n (f n L 2 (S,Λ, and thus, for p > 1, by Lemma 4.1, E[ t I n (f n F t ], ψ 2 E[ t I n (f n F t ] 2 p ψ 2 p where f n (t is the function on R n 1 to C defined by = (n 1!n 2 1 (,t] n 1 f n (t 2 p ψ 2 p (by Proposition 2.4 and Lemma (p 1(n 1 n! f n (t 2 0 ψ 2 p, (4.1 f n (t(t 1,...,t n 1 = f n (t 1,...,t n 1,t. So, for a fixed F (f n L 2 (S,Λ, it follows from (4.1 that lim sup m,m m n=m E[ t I n (f n F t ] 2 p dt lim m In other words, for almost all t R, { m E[ ti n (f n F t ] } m n=m n! f n 2 0 = 0. is a Cauchy sequence in L p, p > 1. Consequently we can define { m } E[ t F F t ] := the L p -limit of E[ t I n (f n F t ] as m. Theorem 4.2. (Clark-Ocone formula for Poisson white noise functionals Let F (f n be in L 2 (S,Λ. Then for p > 5/4, F = E[F]+ te[ t F F t ]dt, in L p, where the above integral is regarded as a Bochner integral. Proof. Letting n (t = {(t 1,...,t n 1 R n 1 ; t j < t, j = 1,2,...,n 1},

10 522 YUH-JIA LEE, NICOLAS PRIVAULT, AND HSIN-HUNG SHIH we have F = E[F]+ = E[F]+ = E[F]+ I n (f n n ( 1 n(t(t 1,...,t n 1 f n (t 1,...,t n 1,tdÑ(t 1 dñ(t n 1 E[ t I n (f n F t ]dñ(t, dñ(t, cf. also Proposition 12 of [15] and Proposition of [16]. Then by applying [10, Theorem 7.2], F = E[F]+ Moreover, since for p > 5/4, 2 2 t E[ ti n (f n F t ] p dt (by Proposition 2.5 t E[ ti n (f n F t ]dt in L. (4.2 δ t p+1/4 E[ t I n (f n F t ] p+1/4 dt 2 (p 5/4(n 1 (n! 1/2 δ t p+1/4 f n (t 0 dt (by 4.1 ( 1/2 ( 2 δ t 2 1 dt 1/2 2 2(p 5/4(n 1 F 0 < +, we see that the identity (4.2 becomes that F = E[F]+ = E[F]+ t E[ ti n (f n F t ]dt t E[ tf F t ]dt in L p, where the above integrals are Bochner integrals.

11 CLARK-OCONE FORMULA OF POISSON FUNCTIONALS 523 Combining Theorem 4.2 with Theorem 3.1 yields that, for any ϕ L 2 (S,Λ, b a d dt Sϕ(P t(gdt ( ( = S t E[ tf F t ]dt (P b (g S = = = b b a S ( t E[ tf F t ] (P b (gdt t E[ tf F t ]dt (P a (g S ( t E[ tf F t ] (P a (gdt S ( t E[ tf F t ] a (gdt S ( t E[ tf F t ] (gdt S ( te[ t F F t ] (gdt, a < b, g L 1 L (R. Therefore, by applying Lebesgue s differentiation theorem, we arrive at the following result, which can be read as the S-transform version of the Clark-Ocone formula. Corollary 4.3. Let ϕ L 2 (S,Λ. Then for any g L 1 L (R, d dt Sϕ(P t(g = S ( t E[ tϕ F t ] (g, [Leb] a.e. t R. References 1. Aase, K., Øksendal, B., Privault, N., Ubøe, J.: White noise generalizations of the Clark- Haussmann-Ocone theorem with application to mathematical finance, Finance Stochast. 4 (2000, Clark, J. M. C.: The representation of functionals of Brownian motion by stochastic integrals, Ann. Math. Stat. 41(1970, de Faria, M., Oliveira, M. J., Streit, L.: A generalized Clark-Ocone formula, Random Oper. and Stoch. Equ. 8 No.2 ( Itô, K.: Spectral type of shift transformations of differential process with stationary increments, Trans. Amer. Math. Soc. 81 (1956, Karatzas, I., Ocone, D., Li, J.: An extension of Clark s formula, Stochastics and Stochastics Reports 37 (1991, Kuo, H.-H.: White Noise Distribution Theory, CRC Press, Lee, Y.J., Shih, H.-H.: The Segal-Bargmann Transform for Lévy Functionals, J. Funct. Anal. 168 ( Lee, Y.J., Shih, H.-H.: Itô formula for generalized Lévy functionals, in: Quantum Information II, Eds. T. Hida and K. Saitô, World Scientific (2000, Lee, Y.J., Shih, H.-H.: The Clark formula of generalized Wiener functionals, in: Quantum Information IV, Eds. T. Hida and K. Saitô, World Scientific, 2002, Lee, Y.J., Shih, H.-H.: Analysis of generalized Lévy white noise functionals, J. Funct. Anal. 211 (2004, Lee, Y.J., Shih, H.-H.: A characterization of generalized Lévy functionals, in: Quantum Information and Complexity, World Scientific, 2005, Nunno, G. D., Øksendal, B., Proske, F.: White noise analysis for Lévy processes, J. Funct. Anal. 206 (2004, Ocone, D.: Malliavin s calculus and stochastic integral representations of functionals of diffusion process, Stochastics. 12 (1984, Parthasarathy, K. R.: An Introduction to Quantum Stochastic Calculus, Birkhäuser, Basel, 1992.

12 524 YUH-JIA LEE, NICOLAS PRIVAULT, AND HSIN-HUNG SHIH 15. Privault, N.: An extension of stochastic calculus to certain non-markovian processes, Prépublication 49, Université d Evry, Privault, N.: Stochastic Analysis in Discrete and Continuous Settings, Lect. Notes in Math., vol. 1982, Springer-Verlag, Berlin, Yuh-Jia Lee: Department of Applied Mathematics, National University of Kaohsiung, Kaohsiung, Taiwan 811 address: yjlee@nuk.edu.tw Nicolas Privault: School of Physical and Mathematical Sciences, Nanyang Technological University, SPMS-MAS, 21 Nanyang Link, Singapore address: nprivault@ntu.edu.sg Hsin-Hung Shih: Department of Applied Mathematics, National University of Kaohsiung, Kaohsiung, Taiwan 811 address: hhshih@nuk.edu.tw